A Catalogue of Orientable 3-Manifolds Triangulated by 30 Colored Tetrahedra

Casali, Maria Rita; Cristofori, Paola

doi:10.1142/s0218216508006312

Cited by 32 publications

(60 citation statements)

References 25 publications

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“…In particular, the generation and classification procedures have been successfully developed in dimension 3, both in the orientable and non-orientable case, allowing the complete topological identification of each involved 3-manifold: see [23] and [12] (resp. [3]) for censuses of orientable (resp.…”

Section: Up-to-date Results From Crystallization Cataloguesmentioning

confidence: 99%

“…Conjectures about significant upper bounds for infinite families of 3-manifolds have also arisen by the analysis of existing crystallization catalogues, which have been generated up to 30 vertices ( [7], [12], [3]). …”

Section: Introductionmentioning

confidence: 99%

“…Actually, a recent result by Swartz ([26]) 12 proves that, for each q, r ≥ 1 odd, k(L(qr + 1, q)) = 2(r + q) − 1; since S(qr + 1, q) = q + r trivially holds ∀r, q, we can state:…”

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confidence: 99%

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A note about complexity of lens spaces

Casali

Cristofori

2014

Forum Mathematicum

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Within crystallization theory, (Matveev's) complexity of a 3-manifold can be estimated by means of the combinatorial notion of GM-complexity. In this paper, we prove that the GM-complexity of any lens space L(p, q), with p ≥ 3, is bounded by S(p, q)−3, where S(p, q) denotes the sum of all partial quotients in the expansion of q p as a regular continued fraction. The above upper bound had been already established with regard to complexity; its sharpness was conjectured by Matveev himself and has been recently proved for some infinite families of lens spaces by Jaco, Rubinstein and Tillmann. As a consequence, infinite classes of 3-manifolds turn out to exist, where complexity and GM-complexity coincide.Moreover, we present and briefly analyze results arising from crystallization catalogues up to order 32, which prompt us to conjecture, for any lens space L(p, q) with p ≥ 3, the following relation: k(L(p, q)) = 5 + 2c(L(p, q)), where c(M ) denotes the complexity of a 3-manifold M and k(M ) + 1 is half the minimum order of a crystallization of M .

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Section: Up-to-date Results From Crystallization Cataloguesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A note about complexity of lens spaces

Casali

Cristofori

2014

Forum Mathematicum

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“…Nevertheless it can be solved for graphs with a low number of vertices by dipole eliminations since, by the 3-dimensional classification results [13], it is known that no rigid crystallization of S 3 exists (different from the "trivial" one of order two) with less than 24 vertices. Furthermore condition (ii) would be very heavy to check, since it implies recognition of 3-spheres with holes.…”

Section: Dimension Fourmentioning

confidence: 99%

“…The usage of supercomputing resources has allowed the generation of catalogues of 3-manifold crystallizations with up to 32 vertices, which have already been completely classified up to 30 vertices (both for orientable [13] and non-orientable [14] cases), and the generation of catalogues of 4-manifold crystallizations up to 20 vertices, for which some classification results have already been obtained from their analysis, as reported in [16].…”

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confidence: 99%

Generation of Catalogues of PL n-manifolds: Computational Aspects on HPC Systems

Marani¹,

Rivi²,

Cristofori³

2013

SCPE

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Abstract. Within mathematical research, Geometric Topology deals with the study of piecewise-linear n-manifolds, i.e. triangulable spaces which appear locally as the n-dimensional Euclidean space. This paper reports on the computational aspects of an algorithm for generating triangulations of PL 3-and 4-manifolds represented by edge-coloured graphs. As the number of graph vertices is increased the algorithm becomes computationally expensive very quickly, making it a natural candidate for the usage of HPC resources. We present an optimized, parallel version of the algorithm that is suitable for deployment of multi-core systems. Scalability results are discussed on two different platforms, namely an IBM iDataPlex Linux cluster and the IBM supercomputer BlueGene/Q.Key words: High Performance Computing, n-manifolds, coloured triangulations, edge-coloured graphs.AMS subject classifications. 57Q15 -57M15 -68W10.1. Introduction. Geometric Topology deals with piecewise-linear (PL) n-manifolds [2], i.e. compact topological manifolds for which there is a triangulation such that each point has a neighbourhood which is piecewise-linearly isomorphic (PL-homeomorphic) to an affine n-simplex. Catalogues of triangulations of PL n-manifolds are valuable sources of data: they yield examples to test conjectures, calculate invariants and make comparisons; they can offer insight into the structural properties of the represented manifolds and may suggest ideas for further theoretical investigation.In particular, since each compact (topological) 3-manifold admits a PL structure and any two PL structures on the same topological 3-manifold are equivalent (i.e. PL-homeomorphic, see [2]), the study of triangulations of PL 3-manifolds is naturally related to the problem of classification, which is still one of the main topics of 3-dimensional topology. The possibility of representing manifolds by combinatorial structures, together with recent advances in computing power, enabled topologists to construct exhaustive tables of small (i.e. obtained by a small number of simplices) 3-manifolds based on different representation methods. In the closed case (i.e. compact and without boundary), catalogues have already been produced and analysed by many authors [8,19,20], with a particular focus on combinatorial properties of minimal triangulations.On the other hand, the problem of classification in dimension four must take into account that a topological 4-manifold not always admits PL structures or may admit non-equivalent ones. For example, although there exists a classification of simply-connected topological 4-manifolds, long established by Freedman [7], the study of (PL) equivalence classes of such structures, especially with regard to their minimal representatives, is an interesting and still open subject of research. Several examples of different PL 4-manifolds triangulating the same topological 4-manifold have recently been presented and the subject is being continuously updated (see for example [9]), but no exhaustive catalogue is available...

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4-colored Graphs and Knot/Link Complements

et al. 2017

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Abstract. A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored graphs (i.e. 4-regular graphs endowed with a proper edge-coloration with four colors) has been recently introduced by two of the authors, and an initial classification of such manifolds has been obtained up to 8 vertices of the representing graphs. Computer experiments show that the number of graphs/manifolds grows very quickly as the number of vertices increases. As a consequence, we have focused on the case of orientable 3-manifolds with toric boundary, which contains the important case of complements of knots and links in the 3-sphere.In this paper we obtain the complete catalogation/classification of these 3-manifolds up to 12 vertices of the associated graphs, showing the diagrams of the involved knots and links. For the particular case of complements of knots, the research has been extended up to 16 vertices.2010 Mathematics Subject Classification: 57M25, 57N10, 57M15.

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A Catalogue of Orientable 3-Manifolds Triangulated by 30 Colored Tetrahedra

Cited by 32 publications

References 25 publications

A note about complexity of lens spaces

A note about complexity of lens spaces

Generation of Catalogues of PL n-manifolds: Computational Aspects on HPC Systems

4-colored Graphs and Knot/Link Complements

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